In 1994, Andrew Wiles solved a mystery that
had stumped mathematicians for 357 years. The 17th-century French mathematician Pierre de
Fermat, states that you can’t find numbers where: x to the n plus y to the n is equal to z to
the n when n is a whole number greater than 2. For example, 3 to the power of
3 + 4 to the power of 3 = 91, which is not the cube of any whole number. Fermat claims to have proved his theorem,
writing in his copy of Arithmetica, an ancient Greek text on number theory: "I have discovered a truly marvelous proof of
this, which this margin is too narrow to contain." Unfortunately, he never wrote down his proof
– not anywhere that anyone has ever found. For over 300 years, mathematicians
puzzled over this perplexing problem, which is similar to the Pythagorean theorem
but applies to exponents greater than 2. Andrew Wiles was a young boy when he stumbled upon Fermat’s Last Theorem at a library
in Cambridge, England in 1963. As he recalled in an interview later featured in the book Fermat’s Enigma by Simon
Singh, which I highly recommend: “Here was a problem that I, a
10-year-old, could understand and I knew from that moment that I would
never let it go. I had to solve it.” His determination was extraordinary considering the brightest minds in math had
failed to crack it for centuries. The 18th-century Swiss mathematician
Leonhard Euler proved a specific case where there are no solutions for n = 3. But
his work didn’t extend to other exponents. Euler even asked a friend to search Fermat’s house just in case he had scribbled down
his proof somewhere, with no luck. The wealthy German industrialist Paul Wolfskehl
was so miserable because the woman he loved didn’t love him back that he planned to end his life,
only to be reinvigorated by Fermat’s puzzle. He never solved it himself but offered a
reward of 100,000 German marks to anyone who could solve it, motivating the
next generation of mathematicians. When Andrew Wiles was a grad student at
the University of Cambridge in the 1970s, his supervisor, John Coates,
suggested he study elliptic curves, which proved crucial for
solving Fermat's Last Theorem. In the 1950s, Japanese mathematicians, Yutaka
Taniyama and Goro Shimura suggested that for every elliptic curve, there is
a corresponding modular form. The Taniyama-Shimura conjecture connected
two distinct areas of mathematics. Later, Ken Ribet, a professor at UC Berkeley,
linked their conjecture to Fermat’s Last Theorem. How? He showed that if Fermat’s
last theorem were false, certain elliptical curves would have to exist. However, the Taniyama-Shimura conjecture
implied that those curves could not exist. Therefore, if the Taniyama-Shimura conjecture were
true, Fermat’s last theorem would also be true. Proving the conjecture would
prove Fermat's Last Theorem. The problem was, Taniyama and Shimura
never proved their own conjecture. Tragically, Taniyama committed suicide
in 1958, leaving a note that read, “I am in the frame of mind that
I lost confidence in my future.” His fiancée took her own life shortly after, writing, “We promised each other
that no matter where we went, we would never be separated. Now that he is
gone, I must go too in order to join him.” In 1986, Andrew Wiles was sipping iced
tea at a friend’s house when his friend mentioned Ken Ribet had proved the link between
Taniyama-Shimura and Fermat’s Last Theorem. “I was electrified. I knew that moment that
the course of my life was changing because this meant that to prove Fermat’s Last Theorem all
I had to do was to prove the Taniyama-Shimura conjecture. It meant that my childhood dream
was now a respectable thing to work on. I just knew that I could never let that go.”
By this time, Andrew had moved to the U.S. where he was a mathematics
professor at Princeton University. When he wasn’t lecturing, he was in his attic
office trying to prove that every single elliptic equation could be correlated with a modular form.
Only his wife knew what he was up to. He kept his work a secret, not wanting
to be distracted, though there may have been another reason for his secrecy.
He may have feared that if his progress leaked, a rival mathematician might beat him
to the punch and claim the glory. He cleverly threw his colleagues off his
track by publishing his ongoing research on elliptic equations every six months, making
it seem like he was keeping busy while actually devoting his time to the age-old enigma.
He began by proving the connection for a specific class of elliptic curves, like
knocking down the first domino in a long line. The challenge was to ensure that if one
domino fell, it would topple the next. Andrew had to show that if one element
of the elliptic curve series matched the corresponding element in the modular form
series, then the next elements would also match. Yet he struggled to topple
the rest of the dominos. Five years after working in secret, he
decided to emerge from his hermit-like state in hopes of finding new ideas.
In 1991, he attended a major math conference in Boston where his Ph.d. supervisor
John Coates mentioned that a student of his, Matthias Flach was working on a powerful
way of analyzing elliptic equations, building on the work of Russian
mathematician Victor Kolyvagin. Andrew realized this Kolyvagin-Flach method
could help him topple the rest of the dominos, ultimately proving every elliptic equation
could be matched to a modular form. But this involved incredibly hard
mathematics and took several more years. After seven years of working all by himself,
Andrew finally decided that he needed some help. In 1993, he confided in his colleague
at Princeton, mathematician Nick Katz. They arranged a secret weekly meeting disguised as
a graduate course where Nick meticulously checked and verified Andrew’s calculations…and assessed
that the Kolyvagin-Flach method worked perfectly. There was still one missing piece of the puzzle. Andrew used a key insight gleaned from a paper by American mathematician Barry Mazur
to complete the puzzle in May 1993. It was time to tell the world that he had
solved the world’s hardest math problem. He chose to do so at a conference
at the Isaac Newton Institute in June 1993 in his hometown of Cambridge.
An air of excitement filled the room as rumors spread of his breakthrough on
the more than 300-year-old mystery. Andrew recalled:
“There was a typical dignified silence while I read out the proof
and then I just wrote up the statement of Fermat’s Last Theorem. I said, ‘I think I’ll stop
here,’ and then there was sustained applause.” Mathematicians usually don't make
headline news, but this was an exception. The excitement was palpable yet
the proof still had to be verified. He submitted his draft to the journal,
Inventiones Mathematicae, where the editor, Barry Mazur,selected six referees
to pore over Andrew’s 200-page work. “...this was subject matter that perhaps only half
a dozen people in the world could fully grasp.” as John Lynch, editor of the award-winning
documentary Fermat’s Last Theorem, put it." During this rigorous scrutiny, a flaw emerged.
One of the referees, Andrew’s Princeton colleague Nick Katz, discovered a gap in the part of the
argument involving the Kolyvagin-Flach method. Nick and Andrew had missed this earlier during
their secret meetings disguised as lectures. Was Andrew’s proof about to
collapse like a house of cards? Despite the setback, he remained optimistic.
He was confident that Kolyvagin-Flach just needed a little tweaking.
He retreated to his attic once again, but it was slow going.
His wife Nada had told him she wanted everything corrected by her
birthday on October 6 — two weeks away. But as time went on with no manuscript
published, rumors began circulating that there were issues with his proof.
Andrew had no choice but to address the speculation publicly in
a forum at the end of 1993: During the review process a number of problems
emerged, most of which have been resolved, but one in particular I have not yet settled.
the calculation of thI believe that I will be able to finish this in the near future using
the ideas explained in my Cambridge lectures. Winter passed and he was still
nowhere close to completing the proof. He decided to bounce ideas off of someone he
trusted: his former student Richard Taylor, now a Cambridge lecturer and one of
the referees who looked over his proof. They worked through the summer
of 1994 without any progress. Andrew was at wit's end and thought of giving up.
But Richard persuaded him to keep working on until Richard returned to Cambridge in September.
The extra month made all the difference. The morning of Monday, September
19, 1994, Andrew was again intensely examining the Kolyvagin-Flach method, trying
desperately to understand why it didn't work. Suddenly, he had an incredible revelation
Although the Kolyvagin-Flach method wasn't entirely successful, it provided enough insight
to revive his original approach using Iwasawa theory - a theory of analyzing elliptic equations
that he had learned as a student at Cambridge. Combining the two methods, which were inadequate
on their own, provided the Eureka moment. It was the most important
moment of my working life. Nothing I ever do again will… I’m sorry. After eight years of dedicating himself to one task, Andrew’s 129-page proof was published
in May 1995, putting to rest the greatest mystery in math. in the Annals of Mathematics
He reflected: “Having solved this problem there’s certainly a
sense of loss, but at the same time there is this tremendous sense of freedom. That particular
odyssey is now over. My mind is at rest.” If you’ve been inspired by Andrew’s
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access to thousands of courses. Thanks for watching. For Newsthink, I’m Cindy Pom.
